Quasi-similarity of contractions having a 2×2 singular characteristic function

Let be a completely non-unitary contraction having a 2×2 singular characteristic function Θ1; that is, Θ1=[θi,j]i,j=1,2 with θij∈H∞ and det(Θ1)=0. As it is well known, Θ1 is a singular matrix if and only if Θ1 can be written as where w1,m1,a1,b1,c1,d1∈H∞ are such that (i) w1 is an outer function with |w1|⩽1, (ii) m1 is an inner function, (iii) 2|a1|+2|b1|=2|c1|+2|d1|=1, and (iv) a1∧b1=c1∧d1=1 (here ∧ stands for the greatest common inner divisor). Now consider a second completely non-unitary contraction having also a 2×2 singular characteristic function . We give necessary and sufficient conditions for T1 and T2 to be quasi-similar.